Sigmahat

Sigmahat, denoted as σ̂, is a notation used to represent an estimator of the population standard deviation σ or, more generally, a model’s scale or dispersion parameter. It is a key quantity for quantifying the variability of data or model residuals and plays a central role in inference, including the construction of confidence intervals and hypothesis tests.

In the context of a normal model with independent and identically distributed observations, σ̂ commonly refers to the estimator of the standard deviation of the data. If the mean is known, the maximum likelihood estimator is σ̂ = sqrt( (1/n) ∑ (x_i − μ)^2 ). If the mean is unknown and estimated from the data (as with the sample mean), a frequently used estimator is σ̂ = sqrt( (1/n) ∑ (x_i − x̄)^2 ). The unbiased estimator for the variance uses the denominator n−1, yielding s^2 = ∑ (x_i − x̄)^2 /(n−1), with s = sqrt(s^2) as the sample standard deviation.

In linear regression and related models, σ̂ denotes the residual standard deviation, computed as σ̂ = sqrt( RSS/(n − p) ), where RSS is the residual sum of squares and p is the number of estimated parameters. This serves as an estimate of the error dispersion and is used to form standard errors for estimated coefficients.

Properties of σ̂ include consistency under standard regularity conditions and sensitivity to departures from model assumptions and to outliers. Robust alternatives exist when dispersion estimation is affected by non-normality or anomalous observations.